Mean square stabilisation of complex oscillatory regimes in nonlinear stochastic systems

References

• Anishchenko, V., Astakhov, V., Neiman, A., Vadivasova, T., & Schimansky-Geier, L. (2007). Nonlinear dynamics of chaotic and stochastic systems: Tutorial and modern developments (Vol. 10). New York, NY: Springer.
   Google Scholar
• Arnold, L. (1998). Random dynamical systems. Berlin: Springer.
   Google Scholar
• Berman, N., & Shaked, U. (2008). H(infinity) control for non-linear stochastic systems: The output-feedback case. International Journal of Control, 81, 1733–1746.
   Web of Science ®Google Scholar
• Chen, G., & Yu, X. (2003). Chaos control: Theory and applications. New York, NY: Springer-Verlag.
   Google Scholar
• Chen, Y., Zheng, W.X., & Xue, A. (2010). A new result on stability analysis for stochastic neutral systems. Automatica, 46, 2100–2104.
   Web of Science ®Google Scholar
• Dharani, S., Rakkiyappan, R., & Cao, J. (2015). Robust stochastic sampled-data H∞ control for a class of mechanical systems with uncertainties. Journal of Dynamic Systems, Measurement, and Control, 137, 101008.
   Web of Science ®Google Scholar
• Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows. Indiana University Mathematics Journal, 2, 193–226.
   Google Scholar
• Fradkov, A.L., & Pogromsky, A.Yu. (1998). Introduction to control of oscillations and chaos. Singapore: World Scientific.
   Google Scholar
• Guo, L., & Wang, H. (2010). Stochastic distribution control system design: A convex optimization approach. New York, NY: Springer-Verlag.
   Google Scholar
• Hirsch, M.W., Pugh, C.C., & Shub, M. (1977). Invariant manifolds. Berlin: Springer-Verlag.
   Google Scholar
• Horsthemke, W., & Lefever, R. (2007). Noise-induced transitions: Theory and applications in physics, chemistry, and biology (Vol. 15). New York, NY: Springer.
   Google Scholar
• Hu, M., Cao, J., & Hu, A. (2014). Mean square exponential stability for discrete-time stochastic switched static neural networks with randomly occurring nonlinearities and stochastic delay. Neurocomputing, 129, 476–481.
   Web of Science ®Google Scholar
• Khasminskii, R.Z. (1980). Stochastic stability of differential equations. Alpen aan den Rijn: Sijthoff & Noordhoff.
   Google Scholar
• Krasnosel'skij, M.A., Lifshits, Je.A., & Sobolev, A.V. (1989). Positive linear systems: The method of positive operators. Berlin: Helderman Verlag.
   Google Scholar
• Krasovskii, N.N., & Lidskii, E.A. (1961). Analytical design of controllers in systems with random attributes. Automation and Remote Control, 22, 1021–1025.
   Google Scholar
• Kurzhanskii, A.B. (1965). On analytic construction of a controller in systems with noise dependent on the control. Differentsial'nye Uravneniya, 1, 204–213.
   Google Scholar
• Kushner, H.J. (1967). Stochastic stability and control. New York, NY: Academic Press.
   Google Scholar
• Levit, M.V., & Iakubovich, V.A. (1972). Algebraic criterion for stochastic stability of linear systems with parametric action of the white noise type. Journal of Applied Mathematics and Mechanics, 36, 130–136.
   Google Scholar
• Mao, X. (1994). Exponential stability of stochastic differential equations. New York, NY: Marcel Dekker.
   Google Scholar
• Mao, X. (1997). Stochastic differential equations and their applications. Chichester: Horwood Publishing.
   Google Scholar
• Riashko, L.B. (1979). Stabilization of linear stochastic systems with state and control dependent perturbations. Journal of Applied Mathematics and Mechanics, 43, 655–663.
   Web of Science ®Google Scholar
• Rodkina, A., & Basin, M.V. (2007). On delay-dependent stability for vector nonlinear stochastic delay-difference equations with Volterra diffusion term. Systems & Control Letters, 56, 423–430.
   Web of Science ®Google Scholar
• Ryashko, L., & Bashkirtseva, I. (2011). Exponential mean square stability analysis of invariant manifolds for nonlinear SDE's. In N. Halidias (Ed.) Stochastic differential equations, series: Mathematics research developments (pp. 67–95). Hauppauge, NY: Nova Science Publishers, Inc.
   Google Scholar
• Ryashko, L.B. (1996). The stability of stochastically perturbed orbital motions. Journal of Applied Mathematics and Mechanics, 60, 579–590.
   Web of Science ®Google Scholar
• Ryashko, L.B. (2007). Exponential mean square stability of stochastically forced invariant manifolds for nonlinear SDE. Stochastics and Dynamics, 7, 389–401.
   Web of Science ®Google Scholar
• Ryashko, L.B., & Shnol, E.E. (2003). On exponentially attracting invariant manifolds of ODEs. Nonlinearity, 16, 147–160.
   Web of Science ®Google Scholar
• Sun, J.-Q. (2006). Stochastic dynamics and control. Amsterdam: Elsevier.
   Google Scholar
• Wiggins, S. (1994). Normally hyperbolic invariant manifolds in dynamical systems. Berlin: Springer-Verlag.
   Google Scholar
• Zhang, J., Liu, Y., & Mu, X. (2015). Further results on global adaptive stabilisation for a class of uncertain stochastic nonlinear systems. International Journal of Control, 88, 441–450.
   Web of Science ®Google Scholar